Topology of Lagrangian fibrations and Hodge theory of hyper-K\"ahler manifolds

Junliang Shen; Qizheng Yin

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arXiv · 1812.10673

Topology of Lagrangian fibrations and Hodge theory of hyper-K\"ahler manifolds

Junliang Shen·Qizheng Yin

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Abstract

We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-K\"ahler manifolds. We present two applications of our result, one on the topology of the base and fibers of a Lagrangian fibration, the other on the refined Gopakumar-Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration associated with a Lagrangian fibration is multiplicative under cup product.

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Junliang Shen, Qizheng Yin. 2018-12-27. Topology of Lagrangian fibrations and Hodge theory of hyper-K\"ahler manifolds. https://arxiv.org/abs/1812.10673

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